Explicit deformation of a spider algebra to a curvilinear scheme via M\"obius generators

Abstract

We construct an explicit flat one-parameter family of 22-dimensional Artinian $k$-algebras whose special fibre is the spider algebra $k[x,y,z]/(x^8, y^8, z^8, xy, xz, yz)$ and whose generic fibre is the curvilinear algebra $k[t]/(t^{22})$. The construction uses M\"obius generators $u_a = t/(1-at)$ inside the curvilinear ring together with a divided-difference change of coordinates, and produces the family via a weighted Rees degeneration with integer coefficients. This gives an explicit one-parameter family witnessing, for this spider ideal, the general phenomenon proved by B\'erczi-Svendsen that every monomial subscheme of $\mathbb{C}^d$ lies in the curvilinear component of the Hilbert scheme of points.